Statistiek I

• Why use statistics?
  • Summarize data (descriptive statistics)
  • Assess relationships in data (inferential statistics)
• Introduction to RStudio and R
  • Variables and functions
  • Import, view and modify data in R
  • Some visualization and statistics in R

• Descriptive statistics vs. inferential statistics
• Sample vs. population
• (Types of) variables
• Distribution of a variable
• Measures of central tendency
• Measures of variation
• Standardized scores
• Checking for a normal distribution

• Descriptive statistics:
  • Statistics used to describe (sample) data without further conclusions
    • Measures of central tendency: Mean, median, mode
    • Measures of variation (or spread): range, IQR, variance, standard deviation
• Inferential statistics:
  • Investigate data of sample in order to infer patterns in the population
    • Statistical tests: \(t\)-test, \(\chi^2\)-test, etc.

• Studying the whole population is (almost always) practically impossible
• Sample is a (selected) subset of population and thus more accessible
  • Selection of representative sample is very important!

• Statistics always involves variables
  • Relations: involve two variables (e.g., English grade vs. English score)
• The values of the variables indicate properties of the cases (i.e. the individuals or entities you study)

• Each case is shown in a row
• Each variable in a column
• For part of our data:

• Nominal (categorical) scale: unordered categories
  • Sex (frequently binary: two categories), Native language, etc.
• Ordinal: ordered (ranked) scale, but amount of difference unclear
  • Rank of English profiency (in class), Likert scale (Rate on a scale from 1 to 5…)

• Interval scale: numerical with meaningful difference but no true 0
  • Year of birth, temperature in Celsius
• Ratio scale: numerical with meaningful difference and true 0
  • Number of questions correct, age
• Scale of variable determines possible statistics
  • E.g., mean age is possible, but not mean native language

table(dat$sex)  # absolute frequencies

# 
#   F   M 
# 346 154

prop.table(table(dat$sex))  # relative frequencies

# 
#     F     M 
# 0.692 0.308

par(mfrow = c(1, 2))
barplot(table(dat$sex), col = c("pink", "lightblue"), main = "abs. frq.")
barplot(prop.table(table(dat$sex)), col = c("pink", "lightblue"), main = "rel. frq.")

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pie(table(dat$study), col = c("red", "cyan", "blue", "yellow"))

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• We are generally not interested in individual values of a variable, but rather all values and their frequency
• This is captured by a distribution
  • Famous distribution: Normal distribution (“bell-shaped” curve): e.g., IQ scores

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• Distribution of a variable shows variability of variable (i.e. frequency of values)

table(dat$english_grade)

# 
#   5 5.5   6 6.5   7 7.5   8 8.5   9 9.5 
#   6   2  76  12 191  21 148  15  26   3

• Histogram shows frequency of all values in groups: \((a,b]\)
  • Look for general pattern, symmetry, outliers

hist(dat$english_grade, xlab = "English grade", main = "")

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• Density curve shows area proportional to the relative frequency

plot(density(dat$english_grade), main = "", xlab = "English grade")

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• The total area under a density curve is equal to 1
• A density curve does not provide information about the frequency of one value
  • E.g., there might be no one who has scored a grade of exactly 6.1
• It only provides information about an interval
  • E.g., more than 50% of the grades lie between 5.5 and 7.5

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• The normal distribution has convenient characteristics
  • Completely symmetric
  • Red area: (about) 68%
  • Red and green area: (about) 95%

• A distribution can also be characterized by measures of center and variation
  • (skewness measures the symmetry of the distribution; not covered in this course)

• Mode: most frequent element (for nominal data: only meaningful measure)
• Median: when data is sorted from small to large, it is the middle value
• Mean: arithmetical average

$$\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{1}{n}\sum\limits_{i=1}^n x_i$$

• (You need to be able to calculate the mean of a series of values by hand, so it is useful to remember the formula)

mean(dat$english_grade)  # arithmetic average

# [1] 7.2828

median(dat$english_grade)

# [1] 7

# no built-in function to get mode: new function
my_mode <- function(x) {
    counts <- table(x)
    as.numeric(names(which(counts == max(counts))))
}
my_mode(dat$english_grade)

# [1] 7

• Quantiles: cutpoints to divide the sorted data in subsets of equal size
  • Quartiles: three cutpoints to divide the data in four equal-sized sets
    • \(q_1\) (1st quartile): cutpoint between 1st and 2nd group
    • \(q_2\) (2nd quartile): cutpoint between 2nd and 3rd group (= median!)
    • \(q_3\) (3rd quartile): cutpoint between 3rd and 4th group
  • Percentiles: divide data in hundred equal-sized subsets
    • \(q_1\) = 25th percentile
    • \(q_2\) (= median) = 50th percentile
    • Score at \(n\)th percentile is better than \(n\)% of scores

quantile(dat$english_grade)  # default: quartiles

#   0%  25%  50%  75% 100% 
#  5.0  7.0  7.0  8.0  9.5

• Minimum, maximum: lowest and highest value
• Range: difference between minimum and maximum
• Interquartile range (IQR): \(q_3\) - \(q_1\)

• A box plot is used to visualize variation of a variable
  • Box (IQR): \(q_1\) (bottom), median (thickest line), \(q_3\) (top)
    • (In example below, \(q_1\) and median have the same value)
  • Whiskers: maximum (top) and minimum (bottom) non-outlier value
  • Circle(s): outliers (> 1.5 IQR distance from box)

boxplot(dat$english_grade, col = "red")

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• Deviation: difference between mean and individual value
• Variance: average squared deviation
  • Squared in order to make negative differences positive
  • Population variance (with \(\mu\) = population mean): $$\sigma^2 = \frac{1}{n}\sum\limits_{i=1}^n (x_i - \mu)^2$$
  • As sample mean ($\bar{x}$) is approximation of population mean ($\mu$), sample variance formula contains division by \(n-1\) (results in slightly higher variance): $$s^2 = \frac{1}{n-1}\sum\limits_{i=1}^n (x_i - \bar{x})^2$$

• Standard deviation is square root of variance $$\sigma = \sqrt{\sigma^2} = \sqrt{\frac{1}{n}\sum\limits_{i=1}^n (x_i - \mu)^2}$$ $$s = \sqrt{s^2} = \sqrt{\frac{1}{n-1}\sum\limits_{i=1}^n (x_i - \bar{x})^2}$$

min(dat$english_grade)  # minimum value

# [1] 5

max(dat$english_grade)  # maximum value

# [1] 9.5

range(dat$english_grade)  # returns minimum and maximum value

# [1] 5.0 9.5

diff(range(dat$english_grade))  # returns difference between min. and max.

# [1] 4.5

IQR(dat$english_grade)  # interquartile range

# [1] 1

var(dat$english_grade)  # sample variance

# [1] 0.75173

sd(dat$english_grade)  # sample standard deviation

# [1] 0.86702

sd(dat$english_grade) == sqrt(var(dat$english_grade))  # std. dev. = sqrt of var.?

# [1] TRUE

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\(P(\mu - \sigma \leq x \leq \mu + \sigma) \approx 68\%\)     (34 + 34)
\(P(\mu - 2\sigma \leq x \leq \mu + 2\sigma) \approx 95\%\)     (34 + 34 + 13.5 + 13.5)
\(P(\mu - 3\sigma \leq x \leq \mu + 3\sigma) \approx 99.7\%\)     (34 + 34 + 13.5 + 13.5 + 2.35 + 2.35)

• It is important to remember these characteristics of the normal distribution!

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\(P(85 \leq \rm{IQ} \leq 115) \approx 68\%\)     (34 + 34)
\(P(70 \leq \rm{IQ} \leq 130) \approx 95\%\)     (34 + 34 + 13.5 + 13.5)
\(P(55 \leq \rm{IQ} \leq 145) \approx 99.7\%\)     (34 + 34 + 13.5 + 13.5 + 2.35 + 2.35)

• IQ scores are normally distributed with mean 100 and standard deviation 15

• Standardization helps facilitate interpretation
• E.g., how to interpret: “Emma’s score is 112” and “David’s score is 105”
• Interpretation should be done with respect to mean \(\mu\) and standard deviation \(\sigma\)
  • Raw scores can be transformed to standardized scores ($z$-scores or $z$-values) $$z = \frac{x - \mu}{\sigma} = \frac{\rm{deviation}}{\rm{standard}\,\rm{deviation}}$$
  • Interpretation: difference of value from mean in number of standard deviations
• (Note that you have to be able to calculate \(z\)-scores!)

• Suppose \(\mu = 108\), \(\sigma = 4\), then: $$z_{112} = \frac{x - \mu}{\sigma} = \frac{112 - 108}{4} = 1$$ $$z_{105} = \frac{105 - 108}{4} = -0.75$$
• \(z\) shows distance from mean in number of standard deviations

• If we transform all raw scores of a variable into \(z\)-scores using: $$z = \frac{x - \mu}{\sigma} = \frac{\rm{deviation}}{\rm{standard}\,\rm{deviation}}$$
• We obtain a new transformed variable whose
  • Mean is 0
  • Standard deviation is 1
• In sum: \(z\)-score = distance from \(\mu\) in \(\sigma\)’s
• \(z\)-scores are useful for interpretation and hypothesis testing (next lecture)

dat$english_grade.z <- scale(dat$english_grade)  # scale: calculates z-scores
mean(dat$english_grade.z)  # should be 0

# [1] 0

sd(dat$english_grade.z)  # should be 1

# [1] 1

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\(P(-1 \leq z \leq 1) \approx 68\%\)     (34 + 34)
\(P(-2 \leq z \leq 2) \approx 95\%\)     (34 + 34 + 13.5 + 13.5)
\(P(-3 \leq z \leq 3) \approx 99.7\%\)     (34 + 34 + 13.5 + 13.5 + 2.35 + 2.35)

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\(P(\mu - \sigma \leq x \leq \mu + \sigma) \approx 68\%\)     (34 + 34)
\(P(\mu - 2\sigma \leq x \leq \mu + 2\sigma) \approx 95\%\)     (34 + 34 + 13.5 + 13.5)
\(P(\mu - 3\sigma \leq x \leq \mu + 3\sigma) \approx 99.7\%\)     (34 + 34 + 13.5 + 13.5 + 2.35 + 2.35)

• If distribution is normal then \(z\)-scores correspond to percentiles
• The function qnorm returns the \(z\)-values for a certain proportion (percentile / 100)

qnorm(95/100)  # z-value associated with 95th percentile

# [1] 1.6449

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• The function pnorm returns the proportion of data < a specified \(z\)-value
  • the percentile can be found by multiplying with 100

100 * pnorm(1.6449)

# [1] 95

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• What proportion of values have a \(z\)-value of at least 1.64?
  • \(P(z \geq 1.64)\)

1 - pnorm(1.64)

# [1] 0.050503

• What proportion of values are located between \(z\)-values between -2 and 2?
  • \(P(-2 \leq z < 2)\)

pnorm(2) - pnorm(-2)

# [1] 0.9545

pnorm(2)

# [1] 0.97725

pnorm(-2)

# [1] 0.02275

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• Some statistical tests require that the data is (roughly) normally distributed
• How to test this?
  • Using visual inspection of a normal quantile plot (or: quantile-quantile plot)
    • A straight line in this graph indicates a (roughly) normal distribution
  • Using the Shapiro-Wilk test (covered in lecture 5)

• Sort the data from smallest to largest to determine quantiles (e.g., percentiles)
  • E.g., median for 50th percentile
• Calculate \(z\)-values belonging to the quantiles (e.g., percentiles) of a standard normal distribution
  • E.g., \(z =\) 0 for 50th percentile, \(z =\) 2 for 97.5th percentile, etc.
• Plot data values ($y$-axis) against normal quantile values ($x$-axis)
  • If points on (or close to) straight line: values normally distributed
• (Note: you need to be able to interpret a quantile-quantile plot, but you don’t have to be able to construct the plot manually)

qqnorm(dat$english_grade)
qqline(dat$english_grade)

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• Distribution not normal for the English grades

qqnorm(dat$english_score)
qqline(dat$english_score)

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• Distribution roughly normal for the English scores

• In this lecture, we’ve covered
  • Descriptive statistics vs. inferential statistics
  • Sample vs. population
  • Four types of variables
  • Distribution of a variable
  • Measures of central tendency
  • Measures of variation
  • Standardized scores
  • How to check for a normal distribution
• In the lab session, you will experiment with descriptive statistics
• Next lecture: Sampling